Trigonometry Examples

$(3 + 2 i) \cdot (x + i y) = i (x - i y) + 4$

Step 1

Simplify $(3 + 2 i) \cdot (x + i y)$ .

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Step 1.1

Rewrite.

$0 + 0 + (3 + 2 i) \cdot (x + i y) = i (x - i y) + 4$

Step 1.2

Simplify by adding zeros.

$(3 + 2 i) \cdot (x + i y) = i (x - i y) + 4$

Step 1.3

Expand $(3 + 2 i) (x + i y)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$3 (x + i y) + 2 i (x + i y) = i (x - i y) + 4$

Step 1.3.2

Apply the distributive property.

$3 x + 3 (i y) + 2 i (x + i y) = i (x - i y) + 4$

Step 1.3.3

Apply the distributive property.

$3 x + 3 (i y) + 2 i x + 2 i (i y) = i (x - i y) + 4$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $2 i (i y)$ .

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Step 1.4.1.1

Raise $i$ to the power of $1$ .

$3 x + 3 i y + 2 i x + 2 (i^{1} i) y = i (x - i y) + 4$

Step 1.4.1.2

Raise $i$ to the power of $1$ .

$3 x + 3 i y + 2 i x + 2 (i^{1} i^{1}) y = i (x - i y) + 4$

Step 1.4.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x + 3 i y + 2 i x + 2 i^{1 + 1} y = i (x - i y) + 4$

Step 1.4.1.4

Add $1$ and $1$ .

$3 x + 3 i y + 2 i x + 2 i^{2} y = i (x - i y) + 4$

Step 1.4.2

Rewrite $i^{2}$ as $- 1$ .

$3 x + 3 i y + 2 i x + 2 \cdot - 1 y = i (x - i y) + 4$

Step 1.4.3

Multiply $2$ by $- 1$ .

$3 x + 3 i y + 2 i x - 2 y = i (x - i y) + 4$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$3 x + 3 i y + 2 i x - 2 y = i x + i (- i y) + 4$

Step 2.2

Multiply $i (- i y)$ .

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Step 2.2.1

Raise $i$ to the power of $1$ .

$3 x + 3 i y + 2 i x - 2 y = i x - (i^{1} i) y + 4$

Step 2.2.2

Raise $i$ to the power of $1$ .

$3 x + 3 i y + 2 i x - 2 y = i x - (i^{1} i^{1}) y + 4$

Step 2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x + 3 i y + 2 i x - 2 y = i x - i^{1 + 1} y + 4$

Step 2.2.4

Add $1$ and $1$ .

$3 x + 3 i y + 2 i x - 2 y = i x - i^{2} y + 4$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite $i^{2}$ as $- 1$ .

$3 x + 3 i y + 2 i x - 2 y = i x - - 1 y + 4$

Step 2.3.2

Multiply $- 1$ by $- 1$ .

$3 x + 3 i y + 2 i x - 2 y = i x + 1 y + 4$

Step 2.3.3

Multiply $y$ by $1$ .

$3 x + 3 i y + 2 i x - 2 y = i x + y + 4$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $i x$ from both sides of the equation.

$3 x + 3 i y + 2 i x - 2 y - i x = y + 4$

Step 3.2

Subtract $i x$ from $2 i x$ .

$3 x + 3 i y + 1 i x - 2 y = y + 4$

Step 3.3

Multiply $i$ by $1$ .

$3 x + 3 i y + i x - 2 y = y + 4$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $3 i y$ from both sides of the equation.

$3 x + i x - 2 y = y + 4 - 3 i y$

Step 4.2

Add $2 y$ to both sides of the equation.

$3 x + i x = y + 4 - 3 i y + 2 y$

Step 4.3

Add $y$ and $2 y$ .

$3 x + i x = 3 y + 4 - 3 i y$

Step 5

Factor $x$ out of $3 x + i x$ .

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Step 5.1

Factor $x$ out of $3 x$ .

$x \cdot 3 + i x = 3 y + 4 - 3 i y$

Step 5.2

Factor $x$ out of $i x$ .

$x \cdot 3 + x i = 3 y + 4 - 3 i y$

Step 5.3

Factor $x$ out of $x \cdot 3 + x i$ .

$x (3 + i) = 3 y + 4 - 3 i y$

Step 6

Divide each term in $x (3 + i) = 3 y + 4 - 3 i y$ by $3 + i$ and simplify.

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Step 6.1

Divide each term in $x (3 + i) = 3 y + 4 - 3 i y$ by $3 + i$ .

$\frac{x (3 + i)}{3 + i} = \frac{3 y}{3 + i} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3 + i$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x (3 + i)}{3 + i} = \frac{3 y}{3 + i} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 y}{3 + i} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify terms.

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Step 6.3.1.1

Simplify each term.

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Step 6.3.1.1.1

Multiply the numerator and denominator of $\frac{3 y}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$x = \frac{3 y}{3 + 1 i} \cdot \frac{3 - i}{3 - i} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2

Multiply.

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Step 6.3.1.1.2.1

Combine.

$x = \frac{3 y (3 - i)}{(3 + 1 i) (3 - i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.2

Simplify the numerator.

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Step 6.3.1.1.2.2.1

Apply the distributive property.

$x = \frac{3 y \cdot 3 + 3 y (- i)}{(3 + 1 i) (3 - i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.2.2

Multiply $3$ by $3$ .

$x = \frac{9 y + 3 y (- i)}{(3 + 1 i) (3 - i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.2.3

Multiply $- 1$ by $3$ .

$x = \frac{9 y - 3 y i}{(3 + 1 i) (3 - i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3

Simplify the denominator.

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Step 6.3.1.1.2.3.1

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

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Step 6.3.1.1.2.3.1.1

Apply the distributive property.

$x = \frac{9 y - 3 y i}{3 (3 - i) + 1 i (3 - i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.1.2

Apply the distributive property.

$x = \frac{9 y - 3 y i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.1.3

Apply the distributive property.

$x = \frac{9 y - 3 y i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2

Simplify.

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Step 6.3.1.1.2.3.2.1

Multiply $3$ by $3$ .

$x = \frac{9 y - 3 y i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.2

Multiply $- 1$ by $3$ .

$x = \frac{9 y - 3 y i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.3

Multiply $3$ by $1$ .

$x = \frac{9 y - 3 y i}{9 - 3 i + 3 i + 1 i (- i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.4

Multiply $- 1$ by $1$ .

$x = \frac{9 y - 3 y i}{9 - 3 i + 3 i - i i} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.5

Raise $i$ to the power of $1$ .

$x = \frac{9 y - 3 y i}{9 - 3 i + 3 i - (i^{1} i)} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.6

Raise $i$ to the power of $1$ .

$x = \frac{9 y - 3 y i}{9 - 3 i + 3 i - (i^{1} i^{1})} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{9 y - 3 y i}{9 - 3 i + 3 i - i^{1 + 1}} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.8

Add $1$ and $1$ .

$x = \frac{9 y - 3 y i}{9 - 3 i + 3 i - i^{2}} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.9

Add $- 3 i$ and $3 i$ .

$x = \frac{9 y - 3 y i}{9 + 0 - i^{2}} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.2.10

Add $9$ and $0$ .

$x = \frac{9 y - 3 y i}{9 - i^{2}} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.3

Simplify each term.

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Step 6.3.1.1.2.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{9 y - 3 y i}{9 - - 1} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.3.2

Multiply $- 1$ by $- 1$ .

$x = \frac{9 y - 3 y i}{9 + 1} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.2.3.4

Add $9$ and $1$ .

$x = \frac{9 y - 3 y i}{10} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.3

Factor $3 y$ out of $9 y - 3 y i$ .

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Step 6.3.1.1.3.1

Factor $3 y$ out of $9 y$ .

$x = \frac{3 y (3) - 3 y i}{10} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.3.2

Factor $3 y$ out of $- 3 y i$ .

$x = \frac{3 y (3) + 3 y (- i)}{10} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.3.3

Factor $3 y$ out of $3 y (3) + 3 y (- i)$ .

$x = \frac{3 y (3 - i)}{10} + \frac{4}{3 + i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.4

Multiply the numerator and denominator of $\frac{4}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$x = \frac{3 y (3 - i)}{10} + \frac{4}{3 + 1 i} \cdot \frac{3 - i}{3 - i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5

Multiply.

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Step 6.3.1.1.5.1

Combine.

$x = \frac{3 y (3 - i)}{10} + \frac{4 (3 - i)}{(3 + 1 i) (3 - i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.2

Simplify the numerator.

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Step 6.3.1.1.5.2.1

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{4 \cdot 3 + 4 (- i)}{(3 + 1 i) (3 - i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.2.2

Multiply $4$ by $3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 + 4 (- i)}{(3 + 1 i) (3 - i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.2.3

Multiply $- 1$ by $4$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{(3 + 1 i) (3 - i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3

Simplify the denominator.

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Step 6.3.1.1.5.3.1

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

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Step 6.3.1.1.5.3.1.1

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{3 (3 - i) + 1 i (3 - i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.1.2

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.1.3

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2

Simplify.

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Step 6.3.1.1.5.3.2.1

Multiply $3$ by $3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.2

Multiply $- 1$ by $3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.3

Multiply $3$ by $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - 3 i + 3 i + 1 i (- i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.4

Multiply $- 1$ by $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - 3 i + 3 i - i i} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.5

Raise $i$ to the power of $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - 3 i + 3 i - (i^{1} i)} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.6

Raise $i$ to the power of $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - 3 i + 3 i - (i^{1} i^{1})} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - 3 i + 3 i - i^{1 + 1}} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.8

Add $1$ and $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - 3 i + 3 i - i^{2}} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.9

Add $- 3 i$ and $3 i$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 + 0 - i^{2}} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.2.10

Add $9$ and $0$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - i^{2}} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.3

Simplify each term.

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Step 6.3.1.1.5.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 - - 1} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.3.2

Multiply $- 1$ by $- 1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{9 + 1} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.5.3.4

Add $9$ and $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{12 - 4 i}{10} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.6

Cancel the common factor of $12 - 4 i$ and $10$ .

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Step 6.3.1.1.6.1

Factor $2$ out of $12$ .

$x = \frac{3 y (3 - i)}{10} + \frac{2 (6) - 4 i}{10} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.6.2

Factor $2$ out of $- 4 i$ .

$x = \frac{3 y (3 - i)}{10} + \frac{2 (6) + 2 (- 2 i)}{10} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.6.3

Factor $2$ out of $2 (6) + 2 (- 2 i)$ .

$x = \frac{3 y (3 - i)}{10} + \frac{2 (6 - 2 i)}{10} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.6.4

Cancel the common factors.

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Step 6.3.1.1.6.4.1

Factor $2$ out of $10$ .

$x = \frac{3 y (3 - i)}{10} + \frac{2 (6 - 2 i)}{2 \cdot 5} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.6.4.2

Cancel the common factor.

$x = \frac{3 y (3 - i)}{10} + \frac{2 (6 - 2 i)}{2 \cdot 5} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.6.4.3

Rewrite the expression.

$x = \frac{3 y (3 - i)}{10} + \frac{6 - 2 i}{5} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.7

Split the fraction $\frac{6 - 2 i}{5}$ into two fractions.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} + \frac{- 2 i}{5} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.8

Move the negative in front of the fraction.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 3 i y}{3 + i}$

Step 6.3.1.1.9

Multiply the numerator and denominator of $\frac{- 3 i y}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 3 i y}{3 + 1 i} \cdot \frac{3 - i}{3 - i}$

Step 6.3.1.1.10

Multiply.

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Step 6.3.1.1.10.1

Combine.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 3 i y (3 - i)}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2

Simplify the numerator.

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Step 6.3.1.1.10.2.1

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 3 i y \cdot 3 - 3 i y (- i)}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.2

Multiply $3$ by $- 3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 i y (- i)}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.3

Multiply $- 3 i y (- i)$ .

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Step 6.3.1.1.10.2.3.1

Multiply $- 1$ by $- 3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y + 3 i y i}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.3.2

Raise $i$ to the power of $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y + 3 y (i^{1} i)}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.3.3

Raise $i$ to the power of $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y + 3 y (i^{1} i^{1})}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y + 3 y i^{1 + 1}}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.3.5

Add $1$ and $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y + 3 y i^{2}}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.4

Simplify each term.

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Step 6.3.1.1.10.2.4.1

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y + 3 y \cdot - 1}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.2.4.2

Multiply $- 1$ by $3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{(3 + 1 i) (3 - i)}$

Step 6.3.1.1.10.3

Simplify the denominator.

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Step 6.3.1.1.10.3.1

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

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Step 6.3.1.1.10.3.1.1

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{3 (3 - i) + 1 i (3 - i)}$

Step 6.3.1.1.10.3.1.2

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Step 6.3.1.1.10.3.1.3

Apply the distributive property.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 6.3.1.1.10.3.2

Simplify.

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Step 6.3.1.1.10.3.2.1

Multiply $3$ by $3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 6.3.1.1.10.3.2.2

Multiply $- 1$ by $3$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Step 6.3.1.1.10.3.2.3

Multiply $3$ by $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - 3 i + 3 i + 1 i (- i)}$

Step 6.3.1.1.10.3.2.4

Multiply $- 1$ by $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - 3 i + 3 i - i i}$

Step 6.3.1.1.10.3.2.5

Raise $i$ to the power of $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - 3 i + 3 i - (i^{1} i)}$

Step 6.3.1.1.10.3.2.6

Raise $i$ to the power of $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - 3 i + 3 i - (i^{1} i^{1})}$

Step 6.3.1.1.10.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - 3 i + 3 i - i^{1 + 1}}$

Step 6.3.1.1.10.3.2.8

Add $1$ and $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - 3 i + 3 i - i^{2}}$

Step 6.3.1.1.10.3.2.9

Add $- 3 i$ and $3 i$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 + 0 - i^{2}}$

Step 6.3.1.1.10.3.2.10

Add $9$ and $0$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - i^{2}}$

Step 6.3.1.1.10.3.3

Simplify each term.

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Step 6.3.1.1.10.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 - - 1}$

Step 6.3.1.1.10.3.3.2

Multiply $- 1$ by $- 1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{9 + 1}$

Step 6.3.1.1.10.3.4

Add $9$ and $1$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{- 9 i y - 3 y}{10}$

Step 6.3.1.1.11

Factor $3 y$ out of $- 9 i y - 3 y$ .

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Step 6.3.1.1.11.1

Factor $3 y$ out of $- 9 i y$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{3 y (- 3 i) - 3 y}{10}$

Step 6.3.1.1.11.2

Factor $3 y$ out of $- 3 y$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{3 y (- 3 i) + 3 y (- 1)}{10}$

Step 6.3.1.1.11.3

Factor $3 y$ out of $3 y (- 3 i) + 3 y (- 1)$ .

$x = \frac{3 y (3 - i)}{10} + \frac{6}{5} - \frac{2 i}{5} + \frac{3 y (- 3 i - 1)}{10}$

Step 6.3.1.2

Combine the numerators over the common denominator.

$x = \frac{3 y (3 - i) + 3 y (- 3 i - 1)}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.3

Simplify each term.

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Step 6.3.1.3.1

Apply the distributive property.

$x = \frac{3 y \cdot 3 + 3 y (- i) + 3 y (- 3 i - 1)}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.3.2

Multiply $3$ by $3$ .

$x = \frac{9 y + 3 y (- i) + 3 y (- 3 i - 1)}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.3.3

Multiply $- 1$ by $3$ .

$x = \frac{9 y - 3 y i + 3 y (- 3 i - 1)}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.3.4

Apply the distributive property.

$x = \frac{9 y - 3 y i + 3 y (- 3 i) + 3 y \cdot - 1}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.3.5

Multiply $- 3$ by $3$ .

$x = \frac{9 y - 3 y i - 9 y i + 3 y \cdot - 1}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.3.6

Multiply $- 1$ by $3$ .

$x = \frac{9 y - 3 y i - 9 y i - 3 y}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.4

Simplify by adding terms.

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Step 6.3.1.4.1

Subtract $3 y$ from $9 y$ .

$x = \frac{- 3 y i - 9 y i + 6 y}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.4.2

Subtract $9 y i$ from $- 3 y i$ .

$x = \frac{- 12 y i + 6 y}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5

Simplify each term.

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Step 6.3.1.5.1

Factor $6 y$ out of $- 12 y i + 6 y$ .

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Step 6.3.1.5.1.1

Factor $6 y$ out of $- 12 y i$ .

$x = \frac{6 y (- 2 i) + 6 y}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5.1.2

Factor $6 y$ out of $6 y$ .

$x = \frac{6 y (- 2 i) + 6 y (1)}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5.1.3

Factor $6 y$ out of $6 y (- 2 i) + 6 y (1)$ .

$x = \frac{6 y (- 2 i + 1)}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5.2

Cancel the common factor of $6$ and $10$ .

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Step 6.3.1.5.2.1

Factor $2$ out of $6 y (- 2 i + 1)$ .

$x = \frac{2 (3 y (- 2 i + 1))}{10} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5.2.2

Cancel the common factors.

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Step 6.3.1.5.2.2.1

Factor $2$ out of $10$ .

$x = \frac{2 (3 y (- 2 i + 1))}{2 (5)} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5.2.2.2

Cancel the common factor.

$x = \frac{2 (3 y (- 2 i + 1))}{2 \cdot 5} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5.2.2.3

Rewrite the expression.

$x = \frac{3 y (- 2 i + 1)}{5} + \frac{6}{5} + \frac{- 2 i}{5}$

Step 6.3.1.5.3

Move the negative in front of the fraction.

$x = \frac{3 y (- 2 i + 1)}{5} + \frac{6}{5} - \frac{2 i}{5}$

Step 6.3.1.6

Combine the numerators over the common denominator.

$x = \frac{3 y (- 2 i + 1) + 6}{5} - \frac{2 i}{5}$

Step 6.3.2

Simplify the numerator.

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Step 6.3.2.1

Factor $3$ out of $3 y (- 2 i + 1) + 6$ .

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Step 6.3.2.1.1

Factor $3$ out of $3 y (- 2 i + 1)$ .

$x = \frac{3 (y (- 2 i + 1)) + 6}{5} - \frac{2 i}{5}$

Step 6.3.2.1.2

Factor $3$ out of $6$ .

$x = \frac{3 (y (- 2 i + 1)) + 3 \cdot 2}{5} - \frac{2 i}{5}$

Step 6.3.2.1.3

Factor $3$ out of $3 (y (- 2 i + 1)) + 3 \cdot 2$ .

$x = \frac{3 (y (- 2 i + 1) + 2)}{5} - \frac{2 i}{5}$

Step 6.3.2.2

Apply the distributive property.

$x = \frac{3 (y (- 2 i) + y \cdot 1 + 2)}{5} - \frac{2 i}{5}$

Step 6.3.2.3

Multiply $y$ by $1$ .

$x = \frac{3 (y (- 2 i) + y + 2)}{5} - \frac{2 i}{5}$

Step 6.3.3

Combine the numerators over the common denominator.

$x = \frac{3 (y (- 2 i) + y + 2) - 2 i}{5}$

Step 6.3.4

Simplify the numerator.

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Step 6.3.4.1

Apply the distributive property.

$x = \frac{3 (y (- 2 i)) + 3 y + 3 \cdot 2 - 2 i}{5}$

Step 6.3.4.2

Simplify.

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Step 6.3.4.2.1

Multiply $- 2$ by $3$ .

$x = \frac{- 6 (y (i)) + 3 y + 3 \cdot 2 - 2 i}{5}$

Step 6.3.4.2.2

Multiply $3$ by $2$ .

$x = \frac{- 6 (y (i)) + 3 y + 6 - 2 i}{5}$

$x = \frac{- 6 y i + 3 y + 6 - 2 i}{5}$

Step 6.3.5

Simplify with factoring out.

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Step 6.3.5.1

Factor $- 1$ out of $- 6 y i$ .

$x = \frac{- (6 y i) + 3 y + 6 - 2 i}{5}$

Step 6.3.5.2

Factor $- 1$ out of $3 y$ .

$x = \frac{- (6 y i) - (- 3 y) + 6 - 2 i}{5}$

Step 6.3.5.3

Factor $- 1$ out of $- (6 y i) - (- 3 y)$ .

$x = \frac{- (6 y i - 3 y) + 6 - 2 i}{5}$

Step 6.3.5.4

Rewrite $6$ as $- 1 (- 6)$ .

$x = \frac{- (6 y i - 3 y) - 1 (- 6) - 2 i}{5}$

Step 6.3.5.5

Factor $- 1$ out of $- (6 y i - 3 y) - 1 (- 6)$ .

$x = \frac{- (6 y i - 3 y - 6) - 2 i}{5}$

Step 6.3.5.6

Factor $- 1$ out of $- 2 i$ .

$x = \frac{- (6 y i - 3 y - 6) - (2 i)}{5}$

Step 6.3.5.7

Factor $- 1$ out of $- (6 y i - 3 y - 6) - (2 i)$ .

$x = \frac{- (6 y i - 3 y - 6 + 2 i)}{5}$

Step 6.3.5.8

Simplify the expression.

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Step 6.3.5.8.1

Rewrite $- (6 y i - 3 y - 6 + 2 i)$ as $- 1 (6 y i - 3 y - 6 + 2 i)$ .

$x = \frac{- 1 (6 y i - 3 y - 6 + 2 i)}{5}$

Step 6.3.5.8.2

Move the negative in front of the fraction.

$x = - \frac{6 y i - 3 y - 6 + 2 i}{5}$